Hi, I need help with the problems 1-5 in my new HW, HW8. Thanks in advance!

Homework Set #8 - Thermodynamic Processes/ Cycles Due: June 22, 2023, 11:59 pm As for all homework sets and examinations: show all of your work. You will be graded on the process of reaching the answer, that is, the solution, and not merely on the final answer. 1. (4 pts) Specific Heat. For this problem you will need to identify whether the gas in question is monatomic, diatomic, or polyatomic (and non-linear). Assume in each case that the "ordinary pressure and ordinary temperature" conditions mean that the kinetic energy of each gas molecule is simply a sum of the translational and rotational energies. a. (1 pt) What is the change in internal energy of 2 moles of nitrogen gas when the temperature rises by 5 K? b. (1 pt) What is the change in internal energy of 3 moles of helium gas when the temperature falls 4 K? c. (1 pt) What is the change in internal energy of 4 moles of methane gas when the temperature rises 3 K? d. (1 pt) What is the change in internal energy of 5 moles of carbon dioxide when the temperature falls 2 K? 2. (6 pts) Pressure. You have confined a sample of air inside a cubical box with each edge length 10 cm at standard temperature and pressure (STP: 0'C, 100 kPa). a. (1 pt) What is the pressure of this gas? b. (1 pt) What force is the gas applying to one face of the cube? c. (1 pt) Making the approximation that each molecule of air has the same component of velocity in the direction toward (or away from) that face of the cube, and that all that the molecules do all day is bounce back and forth between that wall and its opposite partner, what is the impulse (i.e., change in momentum) of a single air molecule bouncing against the wall? Pretend that gas molecules bounce against only the walls, never each other. d. (1 pt) How many times will a single air molecule bounce against that single face in one second? e. (2 pts) Based upon the above approximations, what average force would you predict the gas molecules exert on the single face? 3. (5 pts) Equipartition Predictions. You have a gas of a moles of molecules that each have d degrees of freedom. At a temperature ?: a. (1 pt) What is the total kinetic energy of the gas? b. (1 pt) What is the molar specific heat at constant volume? c. (1 pt) What is the molar specific heat at constant pressure? d. (1 pt) What is the specific heat ratio y = Cp / Cy? e. (1 pt) If d is an integer d23 (since the gas can always translate in three dimensions), what are the minimum and maximum values y can have?4. (2 pts) Adiabatic Processes. What is the ratio of (dP/dl) along an adiabat to (dP/dl ) along an isotherm, at the (P, V,T) point where the two lines intersect? Show your work. 5. (14 pts) Thermodynamic Cycle. Here is a simple Combustion thermodynamic cycle which can convert thermal energy into mechanical energy ("heat into work"). This cycle uses an ideal gas with known specific heats Adiabatic Cy, Cp, and adiabatic ratio y. Assume the (P,V,T) expansion coordinates of each of the numbered states is known. . Starting from state I we compress the gas Cooling adiabatically from the minimum pressure to the Pmin- Adiabatic 4 maximum pressure at state 2. compression Next we burn some fuel which heats the gas min V max isobarically to state 3. Then we adiabatically expand the gas until it has returned to the minimum pressure at state 4. Finally we cool isobarically back to state 1. a. (4 pts) For each of the four processes, calculate the heat added to, or removed from, the engine's working substance. Remember that for proper accounting, take all heat inputs as from one thermal reservoir and all heat outputs as to another reservoir. b. (2 pts) What is the efficiency of this heat engine in terms of the temperatures of the four numbered states? c. (4 pts) Those four temperatures are not independent of one another. They are constrained by the fact that they are each the intersection of an adiabat and an isobar. What is the relationship between the pressures and temperatures of each pair of endpoints for each of the processes? (That is, first for the curve from 1-2, find the relationship between P,, I,, P, and 7, because they are on the same adiabat. Repeat for each process line, recognizing that the relationship between states on a common adiabat and on a common isobar is different.) d. (4 pts) What is the efficiency of this heat engine in terms of the ratio of pressures R= (P./P.) and the adiabatic ratio y? Note that the efficiency and these two ratios are all dimensionless numbers